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This problem, modeled as a two-person zero-sum search game developed on the
lattice
L
is studied in [
9
] . It is called ambush game over time on a cyclic set and
the set of strategies for the hider
X
is equal to
1
n
,
m
F
=
{
A
∈ F
m
:
A
(
i
+
1
)
−
A
(
i
)
∈{
0
,
1
,−
1
,
m
−
1
,
1
−
m
},
n
,
i
=
1
,...,
n
−
1
},
(3.12)
the set of strategies for the searcher is
Y
=
F
n
,
m
and the payoff function
1 f
A
∩
B
= ∅
M
(
A
,
B
)=
.
0 f
A
∩
B
= ∅
Fig. 3.3
Representation of strategy
{
(
1
,
7
)
,
(
2
,
7
)
,
(
3
,
8
)
,
(
4
,
9
)
,
(
5
,
9
)
,
(
6
,
1
)
}
on the lattice
L
=
{
1
,
2
,...,
6
,}×{
1
,
2
,...,
9
}
and on the cyclic set
{
1
,
2
,...,
9
}
Note that the results obtained in Theorems
2
and
3
cannot be applied to this game
because conditions
T
s
X
n
are not satisfied. Now we are going to
define a new transformation that can be applied to make the handling of games
where one of the sets of strategies for the players is the set defined by (
3.12
)ora
subset of it easier. Let
=
X
,
s
=
1
,
2
,...,
T
:
L
−→
L
,
s
=
1
,
2
,...,
n
be the transformation defined by
(
i
,
j
+
1
)
if
j
<
n
,
T
(
i
,
j
)=
(3.13)
(
i
,
1
)
if
j
=
n
.
Figure
3.4
shows the effect of transformation
T
over a subset of
L
=
{
1
,
2
,...,
6
,}×
n
{
1
,
2
,
3
,
4
}
.If
G
=(
X
,
Y
,
M
)
is a game satisfying
X
⊂ F
m
,
TY
=
Y
(or
Y
⊂
,
1
n
,
m
,
TX
F
then Theorem
1
can be applied and
it is easy to prove that there exist optimal mixed strategies for the players such that
=
X
)and
M
(
TA
,
TB
)=
M
(
A
,
B
)
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